Fluctuations for the number of records on subtrees of the Continuum Random Tree

2012/12/21 by Patrick Hoscheit, Hoscheit, Patrick
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1212.5434

openalex publication_date 2012/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the asymptotic behavior af the number of cuts X(T_n) needed to isolate the root in a rooted binary random tree T_n with n leaves. We focus on the case of subtrees of the Continuum Random Tree generated by uniform sampling of leaves. We elaborate on a recent result by Abraham and Delmas, who showed that X(T_n)/√(2n) converges a.s. towards a Rayleigh-distributed random variable Θ, which gives a continuous analog to an earlier result by Janson on conditioned, finite-variance Galton-Watson trees. We prove a convergence in distribution of n^-1/4(X(T_n)-√(2n)Θ) towards a random mixture of Gaussian variables. The proofs use martingale limit theory for random processes defined on the CRT, related to the theory of records of Poisson point processes.

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